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Properties

Label 240.3.bg.e
Level $240$
Weight $3$
Character orbit 240.bg
Analytic conductor $6.540$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,3,Mod(97,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 240.bg (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.53952634465\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 8x^{6} + 269x^{4} - 1116x^{3} + 2312x^{2} + 680x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + ( - \beta_{5} + \beta_{4} - 1) q^{5} + (\beta_{6} + \beta_{5} + \beta_{3} + \cdots - 1) q^{7}+ \cdots + 3 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + ( - \beta_{5} + \beta_{4} - 1) q^{5} + (\beta_{6} + \beta_{5} + \beta_{3} + \cdots - 1) q^{7}+ \cdots + ( - 3 \beta_{6} + 6 \beta_{5} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{5} - 4 q^{7} - 32 q^{11} - 4 q^{13} + 52 q^{17} - 24 q^{21} + 40 q^{23} - 84 q^{25} - 96 q^{31} + 60 q^{33} - 24 q^{35} - 60 q^{37} - 152 q^{41} + 88 q^{43} + 16 q^{47} + 168 q^{51} + 108 q^{53} - 116 q^{55} - 24 q^{57} + 264 q^{61} - 12 q^{63} + 164 q^{65} + 216 q^{67} + 240 q^{71} - 208 q^{73} - 120 q^{75} + 168 q^{77} - 72 q^{81} - 336 q^{83} - 12 q^{85} - 252 q^{87} - 592 q^{91} + 264 q^{93} + 128 q^{95} - 208 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 8x^{6} + 269x^{4} - 1116x^{3} + 2312x^{2} + 680x + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 746 \nu^{7} + 11157 \nu^{6} - 46298 \nu^{5} - 19670 \nu^{4} - 3886 \nu^{3} + 2778413 \nu^{2} + \cdots + 3093010 ) / 1277880 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3096 \nu^{7} - 17905 \nu^{6} + 34598 \nu^{5} + 15150 \nu^{4} + 876904 \nu^{3} - 4737945 \nu^{2} + \cdots + 2940830 ) / 1277880 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2465 \nu^{7} + 10455 \nu^{6} - 21142 \nu^{5} + 2900 \nu^{4} - 663785 \nu^{3} + 2966245 \nu^{2} + \cdots - 838660 ) / 958410 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4775 \nu^{7} - 20987 \nu^{6} + 31148 \nu^{5} - 24410 \nu^{4} + 1286695 \nu^{3} - 5631403 \nu^{2} + \cdots - 227650 ) / 1277880 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 37049 \nu^{7} + 121887 \nu^{6} - 357760 \nu^{5} - 12790 \nu^{4} - 9709721 \nu^{3} + \cdots - 17950270 ) / 3833640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 52139 \nu^{7} + 195804 \nu^{6} - 281506 \nu^{5} + 61340 \nu^{4} - 14348651 \nu^{3} + \cdots - 36387340 ) / 3833640 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21297 \nu^{7} + 76742 \nu^{6} - 171274 \nu^{5} + 62640 \nu^{4} - 5531033 \nu^{3} + \cdots - 821080 ) / 1277880 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -3\beta_{7} - \beta_{6} + 3\beta_{5} - 7\beta_{4} + 4\beta_{3} + \beta_{2} - \beta _1 + 5 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} - \beta_{6} - 7\beta_{5} + 23\beta_{4} + 104\beta_{3} + 21\beta_{2} - \beta _1 + 5 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 17\beta_{7} - 41\beta_{6} - 7\beta_{5} - 17\beta_{4} - 6\beta_{3} - 109\beta_{2} - 51\beta _1 - 45 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{7} - 31\beta_{6} + 3\beta_{5} - 377\beta_{4} + 34\beta_{3} + 411\beta_{2} + 99\beta _1 - 1505 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 587\beta_{7} + 279\beta_{6} - 837\beta_{5} + 1703\beta_{4} + 84\beta_{3} - 279\beta_{2} + 29\beta _1 + 55 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 297 \beta_{7} - 301 \beta_{6} + 1493 \beta_{5} - 7077 \beta_{4} - 22396 \beta_{3} - 6479 \beta_{2} + \cdots - 895 ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4533 \beta_{7} + 8609 \beta_{6} - 457 \beta_{5} + 4533 \beta_{4} + 17294 \beta_{3} + 27541 \beta_{2} + \cdots - 3695 ) / 10 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-\beta_{3}\) \(1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
2.36263 − 2.36263i
−0.137883 + 0.137883i
2.66015 − 2.66015i
−2.88489 + 2.88489i
2.36263 + 2.36263i
−0.137883 − 0.137883i
2.66015 + 2.66015i
−2.88489 − 2.88489i
0 −1.22474 + 1.22474i 0 −3.36263 − 3.70037i 0 8.78825 + 8.78825i 0 − 3.00000i 0
97.2 0 −1.22474 + 1.22474i 0 −0.862117 + 4.92511i 0 −7.33876 − 7.33876i 0 − 3.00000i 0
97.3 0 1.22474 − 1.22474i 0 −3.66015 + 3.40637i 0 2.57407 + 2.57407i 0 − 3.00000i 0
97.4 0 1.22474 − 1.22474i 0 1.88489 − 4.63111i 0 −6.02356 − 6.02356i 0 − 3.00000i 0
193.1 0 −1.22474 − 1.22474i 0 −3.36263 + 3.70037i 0 8.78825 − 8.78825i 0 3.00000i 0
193.2 0 −1.22474 − 1.22474i 0 −0.862117 − 4.92511i 0 −7.33876 + 7.33876i 0 3.00000i 0
193.3 0 1.22474 + 1.22474i 0 −3.66015 − 3.40637i 0 2.57407 − 2.57407i 0 3.00000i 0
193.4 0 1.22474 + 1.22474i 0 1.88489 + 4.63111i 0 −6.02356 + 6.02356i 0 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.bg.e 8
3.b odd 2 1 720.3.bh.o 8
4.b odd 2 1 120.3.u.b ✓ 8
5.b even 2 1 1200.3.bg.q 8
5.c odd 4 1 inner 240.3.bg.e 8
5.c odd 4 1 1200.3.bg.q 8
8.b even 2 1 960.3.bg.k 8
8.d odd 2 1 960.3.bg.l 8
12.b even 2 1 360.3.v.f 8
15.e even 4 1 720.3.bh.o 8
20.d odd 2 1 600.3.u.h 8
20.e even 4 1 120.3.u.b ✓ 8
20.e even 4 1 600.3.u.h 8
40.i odd 4 1 960.3.bg.k 8
40.k even 4 1 960.3.bg.l 8
60.h even 2 1 1800.3.v.t 8
60.l odd 4 1 360.3.v.f 8
60.l odd 4 1 1800.3.v.t 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.u.b ✓ 8 4.b odd 2 1
120.3.u.b ✓ 8 20.e even 4 1
240.3.bg.e 8 1.a even 1 1 trivial
240.3.bg.e 8 5.c odd 4 1 inner
360.3.v.f 8 12.b even 2 1
360.3.v.f 8 60.l odd 4 1
600.3.u.h 8 20.d odd 2 1
600.3.u.h 8 20.e even 4 1
720.3.bh.o 8 3.b odd 2 1
720.3.bh.o 8 15.e even 4 1
960.3.bg.k 8 8.b even 2 1
960.3.bg.k 8 40.i odd 4 1
960.3.bg.l 8 8.d odd 2 1
960.3.bg.l 8 40.k even 4 1
1200.3.bg.q 8 5.b even 2 1
1200.3.bg.q 8 5.c odd 4 1
1800.3.v.t 8 60.h even 2 1
1800.3.v.t 8 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 4T_{7}^{7} + 8T_{7}^{6} + 120T_{7}^{5} + 20900T_{7}^{4} + 120000T_{7}^{3} + 320000T_{7}^{2} - 3200000T_{7} + 16000000 \) acting on \(S_{3}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 12 T^{7} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + \cdots + 16000000 \) Copy content Toggle raw display
$11$ \( (T^{4} + 16 T^{3} + \cdots - 32288)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + \cdots + 53824 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 2570084416 \) Copy content Toggle raw display
$19$ \( T^{8} + 888 T^{6} + \cdots + 82591744 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 3064729600 \) Copy content Toggle raw display
$29$ \( T^{8} + 4500 T^{6} + \cdots + 619810816 \) Copy content Toggle raw display
$31$ \( (T^{4} + 48 T^{3} + \cdots - 1815680)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 16133718222400 \) Copy content Toggle raw display
$41$ \( (T^{4} + 76 T^{3} + \cdots - 63872)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 949580087296 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 287296000000 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 5939943840000 \) Copy content Toggle raw display
$59$ \( T^{8} + 2348 T^{6} + \cdots + 861184 \) Copy content Toggle raw display
$61$ \( (T^{4} - 132 T^{3} + \cdots - 13929728)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 3555366682624 \) Copy content Toggle raw display
$71$ \( (T^{4} - 120 T^{3} + \cdots - 4371968)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 1421016580096 \) Copy content Toggle raw display
$83$ \( T^{8} + 336 T^{7} + \cdots + 2166784 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 693289369600 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
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